Lemma (lcm order)

Let $G$ be a finite abelian group. Let $x,y\in G$ with orders $r,s$. Then $\exists z\in G$ s.t. $$z^{lcm(r,s)}=1$$i.e. $\exists z\in G$ of order $lcm(r,s)$.

Lemma (Lagrange)

Let $\mathbb{F}$ be any field and $f\in\mathbb{F}[x]$ a polynomial of s.t. $\deg f=n$. Then $f$ has at most $n$ roots in $\mathbb{F}$.